# 菲涅耳衍射

$F\ \stackrel{def}{=}\ \frac{a^{2}}{L \lambda}$

## 菲涅耳衍射

$\psi(x,y,z)=-\ \frac{i}{\lambda} \int_{\mathbb{S}} \psi(x',y',0) \frac{e^{ikR}}{R}K(\chi)\ \mathrm{d}x'\mathrm{d}y'$

$K(\chi )=\frac{1}{2}(1+\cos \chi)$

### 菲涅耳近似

$\rho=\sqrt{(x-x')^2+(y-y')^2}$

$(x',y',0)$$(x,y,z)$ 之間的距離 $R$ 可以以泰勒級數表示為

\begin{align}R & =\sqrt{(x-x')^2+(y-y')^2+z^2}=\sqrt{\rho^2+z^2} \\ & = z \sqrt{ 1 + \frac{\rho^2}{z^2} } \\ & = z \left[ 1 + \frac{\rho^2}{2 z^2} - \frac{1}{8} \left( \frac{\rho^2}{z^2} \right)^2 + \cdots \right] \\ & = z + \frac{\rho^2}{2 z} - \frac{\rho^4}{8z^3} + \cdots \\ \end{align}

$\frac{ k \rho^4}{8 z^3} \ll 2 \pi$

$\frac{\rho^4}{8z^3 \lambda} \ll 1$

$\frac{[(x-x')^2+(y-y')^2]^2}{8z^3 \lambda} \ll 1$

$R\approx z + \frac{\rho^2}{2 z} = z + \frac{(x-x')^2 +(y-y')^2}{2 z}$

$z\gg \left(\frac{\rho^4}{8\lambda}\right)^{1/3}=\left[\frac{0.002^4}{8\cdot 500\cdot10^{-9}}\right]^{1/3}\approx 0.016[m]$

### 菲涅耳衍射積分式

 菲涅耳數 $F=a^2/L\lambda$ 菲涅耳衍射區域：$F \ge 1$ 夫朗和斐繞射區域：$F \ll 1$ $a$ － 孔徑或狹縫的尺寸 $\lambda$ － 波長 $L$ － 離開孔徑或狹縫的距離

$\psi(x,y,z)=-\ \frac{ie^{ikz}}{\lambda z} \int_{\mathbb{S}} \psi(x',y',0) e^{ik[(x-x')^2+(y-y')^2]/2z}\ \mathrm{d}x'\mathrm{d}y'$

### 圓孔衍射

$\psi(0,0,z)=-\ \frac{ie^{ikz}\psi_0}{\lambda z} \int_{\mathbb{S}} e^{ik(x'^2+y'^2)/2z}\ \mathrm{d}x'\mathrm{d}y'$

\begin{align}\psi(0,0,z) & =-\ \frac{ie^{ikz} \psi_0 }{\lambda z} \int_0^a e^{ik\rho'^2/2z}\ \rho'\mathrm{d}\rho' \\ & =-\psi_0 e^{ikz} (e^{ika^2/2z}-1) \\ \end{align}

$I(z)=\psi^{*}\psi/2=\psi_0^{\ 2}\ 2\sin^2(ka^2/4z)=I_0\sin^2(ka^2/4z)$

• 極大值：當 $z=\frac{a^2}{2n\lambda},\qquad n=1,2,3,\dots$
• 極小值：當 $z=\frac{a^2}{(2n-1)\lambda},\qquad n=1,2,3,\dots$

$Z_F=\frac{0.001^2}{500\cdot10^{-9}}\approx 2[m]$

[5]

$I=(V0-cos(\frac{u^2+v^2}{2*u}))^2+(V1-sin(\frac{u^2+v^2}{2*u}))^2$

$V_{m}= \sum_{n=0}^\infty *((-1)^n*(\frac{v}{u})^{2*n+m}*J_{2n+m}(v))$

$J_{2n+m}(v)$ 为 第一类$2n+m$贝塞尔函数

### 圆盘衍射

$I=I_0*lambda^2/4$

### 单缝衍射

$I=(Cp(Y)-Cq(Y))^2+(Sp(Y)-Sq(Y))^2$

$Cp(Y) := \int_0^p(\cos((1/2)*\pi*t^2)\,dt$

$Cq(Y) =\int_0^q (\cos((1/2)*\pi*t^2)\,dt$;

Sp,Sq 为正弦菲涅耳积分：

$Sp(Y) := \int_0^p(\sin((1/2)*\pi*t^2)\,dt$

$Sq(Y) =\int_0^q (\sin((1/2)*\pi*t^2)\,dt$;

### 直边衍射

$I=(Cp(Y)+0.5)^2+(Sp(Y)+0.5))^2$

$Cq(Y) =\int_0^q(\cos((1/2)*\pi*t^2)\,dt$;

Sp 为正弦菲涅耳积分：

$Sp(Y) := \int_0^p(\sin((1/2)*\pi*t^2)\,dt$

## 進階理論

### 卷積

$h(x,y,z) =-\ \frac{ie^{ikz}}{\lambda z} e^{i \frac{k}{2 z} (x^2 + y^2)}$

$\psi(x,y,z)=\iint\limits_{-\infty}^{\ \ \ \infty} \psi(x',y',0) h(x-x',y-y',z) \ \mathrm{d}x'\mathrm{d}y'$

$\psi_z(x,y)=\iint\limits_{-\infty}^{\ \ \ \infty} \psi_0(x',y') h_z(x-x',y-y') \ \mathrm{d}x'\mathrm{d}y'$

$\psi_z(x,y)=\psi_0(x,y)*h_z(x,y)$

$\mathcal{F}\{\psi_z(x,y)\}=\mathcal{F}\{\psi_0(x,y)*h(x,y)\}=\mathcal{F}\{\psi_0(x,y)\}\cdot\mathcal{F}\{h_z(x,y)\}$

$G(X,Y)=\mathcal{L}\{f(x,y)\}$

$G(X,Y)=\mathcal{L}\left\{\iint\limits_{-\infty}^{\ \ \ \infty} f(x',y') \delta(x-x')\delta(y-y') \ \mathrm{d}x'\mathrm{d}y' \right\}$

$f(x',y')$ 視為函數 $\delta(x-x')\delta(y-y')$ 權重係數，應用線性系統的性質，可以將積分式寫為

$G(X,Y)=\iint\limits_{-\infty}^{\ \ \ \infty} f(x',y') \mathcal{L}\{\delta(x-x')\delta(y-y')\} \ \mathrm{d}x'\mathrm{d}y'$

### 傅立葉變換

$K_x\ \stackrel{def}{=}\ kx/z$
$K_y\ \stackrel{def}{=}\ kx/z$

$(x-x')^2 = x^2 + x'^2 -2 x x'$
$(y-y')^2 = y^2 + y'^2 -2 y y'$

$G(K_x,K_y)\ \stackrel{def}{=}\ \mathcal{F} \left\{ g(x,y) \right\}\ \stackrel{def}{=}\ \iint\limits_{-\infty}^{\ \ \ \infty} g(x',y') e^{-i (K_x x' + K_y y')}\ \mathrm{d}x'\mathrm{d}y'$

$g(x',y')=\psi_0(x',y') e^{ik(x'^2+ y'^2)/2z}$

\begin{align}\psi_z(x,y) & =-\ \frac{ie^{ikz}}{\lambda z} e^{ik(x^2+y^2)/2z}\ \mathcal{F}\{ \psi_0(x',y') e^{ik(x'^2+y'^2)/2z}\} \\ & =-\ \frac{ie^{ikz}}{\lambda z} e^{ik(x^2+y^2)/2z}\ \mathcal{F}\{ g(x',y')\} \\ & =-\ \frac{ie^{ikz}}{\lambda z} e^{ik(x^2+y^2)/2z}\ G(K_x,K_y) \\ & =h_z(x,y)\ G(K_x,K_y) \\ \end{align}

## 參考文獻

1. ^ M. Born & E. Wolf, Principles of Optics, 1999, Cambridge University Press, Cambridge
2. ^ 實際而言，在先前一個步驟裏做了一個近似，即假定 $e^{i k r}/r$ 是真實波，但這不是向量亥姆霍茲方程式的解答，而是純量亥姆霍茲方程式的解答。請參閱條目純量波近似（scalar wave approximation）。
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• Goodman, Joseph W. Introduction to Fourier optics. New York: McGraw-Hill. 1996. ISBN 0-07-024254-2.